Is atheism a belief system, a new 'faith' as it were, in the same way that atheists claim that theism is a belief system?

It's less of a belief system and more of a cult that attempts to prey on believers to strip them of their belief. It's pretty much proven those atheists are a sad and sorry lot and engage in the "misery loves company" dynamic. I think there is enough research that supports this conclusively.

Atheism isn't a belief system. Atheism is a belief -- one view on one issue.

Atheism is to religion what a non-plumber is to a plumber. There should not actually be a word needed to describe a non something. Non belief is the default after all. No one is born a certain religion. Just as no one is born a plumber.
Atheism has no gods, no magic, no belief in the supernatural.
Is it a belief yes. But so is the belief that the floor exists. But is it a theistic faith? No.
Theists try to bring atheistism down to their level so they can strawman it and then use whataboutism. But even if they succeed in doing that that still doesn't get rid of the arguments against theism.

No

But some can be annoying in the same way

A belief system just as much as conservativ[b]ism[/b], liberal[b]ism[/b] commun[b]ism[/b] or intellectual[b]ism[/b]. All are beliefs.

If you wish to add "faith" though, then it becomes a religion. Otherwise all the above are also religions. Can you really call all those religions?

No it’s an absence of one.

Atheism only describes a non-belief in organized religion and the supernatural. That label does not describe what a person believes, only what they dont believe in.

I call myself a secular humanist, not an atheist.

Not exactly. There are four classes of belief, and three of them are atheistic. The skeptics you allude to do have a belief. It's that nature denies afterlife of any sort.

It’s definitely a belief system but I wouldn’t call it faith.

No. lol